Deterministic Polarization Entanglement Purification of W State in Multiple Degrees of Freedom with Hyper-Entanglement

We present two deterministic entanglement purification protocols for distilling W state in polarization entanglement in a one-step fashion. The three parties can obtain a maximally entangled W polarization state with success probability 100 % from the hyper-entangled W state in multiple degrees of freedom (DOFs), and they do not require the quantum system to be entangled in polarization DOF but in spatial DOF. The former protocol is implemented with the passive linear optics, which is feasible with current experiments. The later one is implemented with cross-Kerr-nonlinearities, which can achieve higher efficiency of the entanglement transformation among different DOFs since it does not require the sophisticated single-photon detector. The present protocols are both suitable for polarization entanglement purification of multi-photon-entangled W state in quantum computation network because they need less operations and classical communications in the practical implementation.